Decay and Scattering of Small Solutions of a Generalized Boussinesq Equation

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Journal of Functional Analysis � FU3052

journal of functional analysis 147, 51�68 (1997)

Decay and Scattering of Small Solutions of aGeneralized Boussinesq Equation

Yue Liu*

Department of Mathematics, The University of Texas at Austin, Austin, Texas 78712

Received December 18, 1995; revised May 2, 1996; accepted September 4, 1996

We study the long-time behavior of small solutions of the initial-value problemfor a generalized Boussinesq equation. We obtain a lower bound for the degrees ofnonlinearity which allows us to establish a nonlinear scattering result for small per-turbations; that is, the small solutions of the nonlinear problem behave asymptoti-cally like the solution of the associated linear problem. Under certain hypotheses,we can construct a scattering operator for the Boussinesq equation which carries aneighborhood of 0 in the energy space X into X. � 1997 Academic Press

1. INTRODUCTION

In this paper, we mainly study the long-time behavior of solutions andnonlinear scattering theory for a generalized Boussinesq equation

utt&uxx+(uxx+ f (u))xx=0, for x # R (BQ)

or, equivalently, the system of equations

{ut=vx ,vt=(u&uxx& f (i))x ,

for x # R.

In the theory of water waves, there is a competition between the effectsof nonlinearity and dispersion. Such problems are modeled by equationssuch as Korteweg-de Vries equation (KdV) and Boussinesq equation[Bou, Ca]. This generalized Boussinesq equation (BQ) is also a model inthe theory of phase transitions in shape-memory alloys [FaLaSp] and inthe study of anharmonic lattice waves [SmCh]. Like (KdV), (BQ) has thesolitary wave solutions for certain nonlinearity f. In fact, for f (u)=|u| p&1 u

article no. FU963052

510022-1236�97 �25.00

Copyright � 1997 by Academic PressAll rights of reproduction in any form reserved.

* I thank the reviewer for some valuable comments.


with p>1, explicit formulas for the solitary wave solutions are known. If|c|<1, we have the solitary wave .� c=(.c , &c.c) of (BQ),

.c(!)=s1 sech2�p&1(s2!), (1.1)

where

s1=s1(c, p)=( 12 ( p+1)(1&c2))1�( p&1)

and

s2=s2(c, p)= 12 ( p&1)(1&c2)1�2

with !=x&ct. The existence of solitary waves of (BQ) has been attributedto a balance between the effects of the nonlinearity and dispersive terms.However, for certain choices of the nonlinear term f, there exist solutionsof (BQ) in which dispersive effects seem to dominate. For (KdV)

ut+uxxx+( f (u))x=0,

it has been shown [Str1, Kl, Sh, KIPo, PoVe, ChWe]. It is known that iff ({)=(|{| p) as { � 0, then for some large enough power p> p0 , solutionsof (KdV) with sufficiently small initial data decay to zero, uniformly in xas t � � and that small solution of behave asymptotically likes the solu-tion of the associated linear problem. In fact, the lower bound to p0=(9+- 73)�4r4.39 was obtained by Ponce and Vega [PoVe], and it wasimproved by Christ and Weinstein [ChWe] to p0=(23&- 57)�4r3.86.Similar result was obtained in [Al] for Benjamin�Bona�Mahoney equa-tion (BBM). Our purpose is to show if the initial data is sufficiently smallin some suitable norms, then the solution of (BQ) decays uniformly to zeroas t � �. More specifically, if f ({)=O( |{| p), as { � 0, dispersion occurs(Remark of Theorem 1.1) provided

p> p0#2+- 7r4.64575 (1.2)

and we look for a solution u� \ of free Boussinesq equation

{ut=vx ,vt=(u&uxx)x ,

for x # R (1.3)

such that the difference of solution u� of (BQ) and u� \ tends to zero in theenergy space as t � \� (Theorem 1.2). The suitable regularity hypotheseson f and the initial data u� (0)=u� (0)=u� 0 guarantee the global existence ofa unique solution u� of (BQ) for all time t # R (Theorem 3.1). The methodof the dispersion proof is based on a modification of Strauss [Str1], Christand Weinstein [CrWe], and Ponce and Vega [PoVe]. That is, we write

52 YUE LIU


(BQ) as an integral equation and treat the nonlinearity as a small pertur-bation of the linear part of the equation. We then utilize an estimate for theuniform decay of solutions of linearized version of (1.3) to obtain a prioriestimates on time-weighted norms of solution u� of (BQ).

Our second purpose is to construct scattering operator S. The scatteringoperator is defined as S( g� &)= g� +, where S(t) g� \=u� \(t) satisfies

&u� (t)&u� \(t)&X � 0, as t � \�, (1.4)

where u� (t) is the solution of (BQ), u� \=S(t) g� \ is the solution of (1.3)with the initial data g� \ and X is the energy space.

The scattering theory was initialed by Segal [Se1, Se2] for nonlinearwave equation (NLW) and Strauss [Str2, Str3] in studying nonlinearSchro� dinger equation (NLS), nonlinear Klein-Gordon equation (NLKG)and generalized Korteweg-de Vries equation (GKdV). It is also inspired byGinibre and Velo [GiVe] for (NLS) and Brenner [Br] for (NLKG). Inthese versions S is only defined on some certain dense subsets. Morawetzand Strauss [MoSt] proved for (NLKG) that S: 7 � 7 for a certain denseclass 7 for n=3. This was extended by Lin and Strauss [LiSt] to (NLS).Strauss [Str4] proved that S: N � X for some neighborhood N of 0 # Xfor both the (NLS) and (NLKG) cases. I want to construct S for someneighborhood N of 0 # X for (BQ) (Theorem 4.1). The method of proofbasically follows the ideas of Strauss [Str4]. That is, write (BQ) as theintegral form, treating the integral equation as a map in the some spaces.We then use an estimate for decay of solutions in the some norms oflinearized version (1.3) to obtain the contraction map.

We will use the following notations. The subscripts to denote partialderivatives. Let u� (t)=(u(t), v(t)) denote the vector function x [ u� (x, t).Let |u� (t)|p, q# |u(t)|p+|v(t)|q denote the norm in L p_Lq, for 1� p,q�+�, | } |p refers the L p#(L p(R, dx) norm. Let &u� (t)&s, p#&u(t)&s, p+&v(t)&s&1, p for real s, 0< p�+�, denote the norm of the space X p

s #L p

s _L ps&1 , where L p

s =J&sL p denotes the Bessel potential space with thepotential Js=(1&�2

x)s�2 whose norm will be denoted by & }&s, p=&Js }&p .When p=2 we will write Hs instead of L2

s , with norm & }&s instead of & }&s, 2

and Xs=X 2s =Hs_H s&1 with the norm &u� &s=&(u, v)&s=&u&s+&v&s&1 .

Now we state our results on the dispersion and scattering of low-energyfor (BQ). We obtain:

Theorem 1.1. Let p>5, and f # C 1(R) satisfy | f ({)|=O( |{| p) and| f $({)|=O( |\| p&1) as { � 0. Then there exists $>0 such that for any u� 0=(u0 , v0) # X1 & X 1

0 satisfying

|u0| 1+|J&1v0| 1+&u0&1+|v0| 2<$,

53A GENERALIZED BOUSSINESQ EQUATION


the solution u� of (BQ) with u� (0)=u� o is in C(R; X1) satisfying

supt # R

(1+|t| )1�3 |u(t)|�<�. (1.5)

Remark. By some Sobolev's embeddings, one can easily improveTheorem 1.1 as follows:

Let p> p0#2+- 7 and f # C 1(R) satisfy f ({)=O( |{| p) and f $({)=(|{| p&1) as { � 0. Then there exists $>0 such that for any 1�( p+1)<k� 1

2 , s+k>1, and u� 0 # X 1k+s+1�2 satisfying

&u� 0&{(k, s), q<$;

where {(k, s) = ( p & 1)�( p + 1)(k + s + 1�2) + (2�( p + 1))k and 1�q +1�( p+1)=1, the solution u� of (BQ) with u� (0)=u� 0 in C(R; X1) satisfying

supt # R

(1+|t| )1�3(1&2�p+1) |u(t)|�<�. (1.6)

Moreover,

supt # R

(1+|t| )1�3(1&2�p+1) &u� (t)&k, p+1<+�. (1.7)

Theorem 1.2 (scattering). Let u� (t) be the solution of (BQ) in Theorem1.1. Then there is a unique solution u� \(t) of the linearized Eq. (1.3) such that

&u� (t)&u� \(t)&1 � 0, as t � \�. (1.8)

Remark 1. If p> p0=2+- 7, then one can obtain

&u� (t)&u� \(t)&k, p+1 � 0, as t � \�, (1.9)

and

&u� (t)&u� \(t)&: � 0, as t � \�, (1.10)

where 1�( p+1)<k� 12 and : # [0, 1].

Remark 2. If f (u)=|u| p&1 u and |c|<1, (BQ) has the solitary wavesolutions .� c=(.c , �c) which is explicitly expressed in (1.1), where .c(!) isan exponentially decaying function of !, but is not decaying with time.A simple calculation shows that

&u� 0&0, 1+&u� 0&1 � 0, as |c| � 1

for 1< p<3. Therefore, the conclusions of Theorem 1.1 and Theorem 1.2are not valid for 1< p<3. We do not know what the best value of p mightbe.

54 YUE LIU


The arguments we use are based on estimates for the free group ofunitary operators S(t) for the linearized Eq. (1.2). That is, we treat the non-linearity as a perturbation and write (BQ) as

u� (t)=S(t)u� 0+|t

0S(t&{) f9 (u� ({)){,

where f9 (u� )=(0, �x f (u)).In Section 2, we derive the required estimates for the one parameter group

of transformations u� 0 [ S(t)u� 0 . We prove Theorem 1.1 and Theorem 1.2 inSection 3. In Section 4, we will construct the scattering operators S.

2. ESTIMATES FOR THE LINEARIZED EQUATION

In this section, we are going to deduce L1 � L� estimate and L p � Lq

estimate.

Lemma 2.1 (L1 � L�). Let S(t) be a group of unitary operators for thelinearized Eq. (1.3), that is, u� (t)=S(t)u� 0=(u(t), v(t)) is the solution of

u� t+Au� x=0 (2.1)

with u� (0)=u� 0=(u0 , v0), where A=( 0&J 2

&10 ). If u� 0 # X 1

k for k # R, thenu� (t) # X �

k and satisfies

&u� (t)&k, ��C( |t| &1�3+|t|&1�2) &u� 0&k, 1 , for t{0. (2.2)

If u� 0 # Xs0+k & X 1k for some s>1�2 and k # R, then

&u� (t)&k, ��C(1+|t| )&1�3 (&u� 0&k+s0+&u� 0&k, 1). (2.3)

If u0 # Xs+k+1�2, 1 , for some s>1�2, and s+k>1, then

&u� (t)&k, ��C(1+|t| )&1�3 &u� 0&k+s+1�2, 1 . (2.4)

In order to prove Lemma 2.1, we need the following lemma which wasalready proved in [Liu]. The following lemma is an application of thewell-known Van der Corput lemma.

Lemma 2.2. For t{0 we have

sup: # R } |

�

&�ei th(!,:) d!}�C( |t| &1�2+|t|&1�2), (2.5)

where h(!, :)=! - 1+!2+:!, and C is a constant.



Proof. See Lemma 1.6 [Liu].

Proof of Lemma 2.1. We can write the solution u� (t)=S(t)u� 0 of (2.1) as

u� (t)=|�

&�eix! \ cos(t!(!) )

i(!) sin(t!(!) )i(!) &1 sin(t!(!) )

cos(t!(!) ) + @u� 0(!) d!, (2.6)

where u�^ 0 is the Fourier transform of u� 0 and (!) =(1+|!| 2)1�2. Since

Jku� (t)=S(t) J ku� 0

=|�

&�eit! \ cos(t!(!) )

i(!) sin(t!(!) )i(!) &1 sin(t!(!) )

cos(t!(!) ) + J ku� 0@(!) d! (2.7)

it follows that

&u� (t)&k, ��C7 } |�

&�(Jku0@\(!) &1 Jkv0@) eit(!(!)\x!�t) d!}

�C7 |�

&�|J ku0( y)\J k&1v0( y)| dy } |

�

&�eit(!(!)\x!�t) d!} , (2.8)

where the sum 7 are over all two sign combinations. Using Lemma 2.2, weobtain (2.2), that is,

&u� (t)&k, ��C( |t|&1�2+|t|&1�3)( |Jku0| 1+|Jk&1v0| 1)

�C( |t|&1�2+|t| &1�3) &u� 0&k, 1 .

In order to show (2.3), we estimate from (2.7)

&u� (t)&k, ��C |�

&�( |Jku0@(!)|+|J k&1@v0(!)| ) d!

�C \|�

&�(1+|!| 2)&s0 d!+

1�2

( |(!) k+s0 u0| 2+|(!) k+s0&1 v0| 2)

�C(&u0&k+s0+&v0&k+s0&1)�C &u� 0&k+s0

(2.9)

for s0>1�2. Combining (2.9) with the estimate (2.2), we obtain (2.3). Sincelinear operator S(t) is unitary in Xk for any k # R, it follows that

&u� (t)&k=&S(t)u� 0&k=&u� 0&k . (2.10)

By (2.3), we obtain

&u� (t)&k, ��C(1+|t| )&1�3 &u� 0&k+s+1�2, 1 . (2.11)

56 YUE LIU


This estimate can be done by using Sobolev's embedding theoremL1

k+s+1�2+i � L2k+s0+i for s>s0 , s0+k�1 and i=0, 1. Applying the inter-

polation theorem for evolution operator S(t) to (2.2), (2.4) and (2.10), weobtain:

Lemma 2.3 (L p � Lq). If u� 0 # X 1k & Xk , for some real k, then

u� (t) # X p+1k and satisfies

&u� (t)&k, p+1=&S(t)u� 0&k, p+1�C( |t|&1�2+|t| &1�3)(1&2�( p+1)) &u� 0&k, q . (2.12)

If u� 0 # X 1k+s+1�2 for some s>1�2 and k+s>1, then

&u� (t)&k, p+1�C(1+|t| )&1�3(1&2�( p+1)) &u� 0&(1&%)(s+k+1�2)+%k, q

=C(1+|t| )&1�3(1&2�( p+1)) &u� 0&{(s,k), q , (2.13)

where 1�q+1�( p+1)=1 and %=2�( p+1).

3. PROOF OF THEOREMS

In this section we are going to prove Theorem 1.1 and Theorem 1.2.First of all, we give a simpler proof of the global existence of small solutionfor (BQ), which was first proved by Linares [Li].

Theorem 3.1. Let f # C 1(R) such that | f ({)|=O( | | p) with p>1 as � 0.Then, there is $>0, such that if &u� 0&1<$, (BQ) has a unique solutionu� # C(R; X1) with u� (0)=u� 0 . Moreover, &u� (t)&1�C &u� 0&1 , for all time t # Rand the energy E and the momentum Q are independent of t, that is,

E(u� (t))=|�

&�( 1

2u2+ 12u2

x+ 12v2&F(u)) dx=E(u� 0)

and

Q(u� (t))=|�

&�uv dx=Q(u� 0),

where F $= f and F(0)=0, and the constant C only depends on &u� 0&1 .

To prove Theorem 3.1, we need the following local existence for (BQ)which was proved by Liu [Liu].

Lemma 3.2. Let u� 0=(u0 , v0) # X1=H 1_L2. If f # C 1(R) with f (0)=0,then there exist T>0 and a unique solution u� (t)=(u(t), v(t)) of (BQ) in



C([0, T ); X1) with u� 0=u� 0 and E and Q are independent of t. Moreover, ifT<�, then

lim supt � T&

&u� (t)&1=�

Remark. Since (BQ) will not change when t is switched to &t, the solu-tion u� (t) in Lemma 3.2 can be extended to u� # C((&T, T )); X1).

Proof of Theorem 3.1. Without loss of generality we restrict ourselves tothe case t>0. By Lemma 3.2, it suffices to prove that &u� (t)&1 is boundedin [0, T ). In fact, using the conserved energy E(u� (t))=E(u� 0) we obtain

&u� (t)&21�C |E(u� 0)|+C |

�

&�|u| p+1 dx

�C(&u� 0&21+&u(t)& p+1

1 ) for t # [0, T ), (3.1)

where the constant C depends only on &u� 0&1 .Define M(t)=sup0�<t &u� ( )&X1

. Then by (2.1) we have

M(t)�C$+CM(t):, (3.2)

where :=( p+1)�2>1. Hence, for sufficiently small $ such that &u� 0&1<$,it follows from the continuity of M(t) that M(t) remains in the boundedconnected component of [ y�0; y�C$+Cya] containing the origin forall t # [0, T ). Moreover, we have M(t)�C &u� 0&1 for all time t # [0, T).

Proof of Theorem 1.1. We write (BQ) as the integral equation

u� (t)=S(t)u� 0+|t

0S(t&{) f9 (u� ({)) d{, (3.3)

where f9 (u� )=(0, �x f (u)). By Lemma 2.1 and Theorem 3.1, we estimate fork=0,

|u(t)|��C(1+|t| )&1�3 (&u� 0&s+&u� 0&0, 1)

+|t

0( |t&{|&1�2+|t&{|&1�3) |( f (u))x | 1 d{

�C(1+|t| )&1�3 $+C |t

0( |t&{|&1�2+|t&{|&1�3)

_| f $(u) u&1| � |u| 2 |ux | 2 d{

�C(1+|t| )&1�3 $+C$ |t

0( |t&{|&1�2+|t&{|&1�3) |u({)| p&2

� d{.

58 YUE LIU


To formulate the main estimate, we introduce the notation

M�(t)= sup0�{<t

(1+{)1�3 |u({)|� .

Without loss of generality we restrict ourselves to the case t>0. Byestimate (3.4), we obtain

M�(t)�C$(1+M p&2� (t) I(t)), (3.5)

where

I(t)=(1+t)1�3 |t

0( |t&{|&1�2+|t&{|&1�3)(1+{)&( p&2)�3 d{. (3.6)

Since p>5, that is ( p&2)�3>1, the integral I(t) is bounded for all t�0.Hence, we have

M�(t)�C$(1+M p&2� (t)). (3.7)

By u� (t) # C(R; X1), we have u(t) # C(R; L�). Therefore, if $ is chosensufficiently small, M�(t) is bounded for all time t. This completes the proofof Theorem 1.1.

In order to show Theorem 1.2, we need the following inequality ofGagliardo�Nirenberg type.

Lemma 3.3. If f # L p0s0

& L p1s1

with p0 , p1 # (1, �) and s0 , s1 # R, then

& f &s, p�C & f &%s0, p0

& f &1&%s1, p1

,

where s=%s0+(1%) s1 and 1�p=%�p0+(1&%)�p&1.

Proof. See [BeLo, Chap. 4].

The proof of Theorem 1.2. Let u� (x, t) be the solution of (BQ) inTheorem 1.1. We define

u� \(t)=u� (t)+|\�

tS(t&{) f9 (u� ({)) d{. (3.8)

Now we only consider the case of u� +, since the proof of u� & is similar.If u� (t) is the solution in Theorem 1.1, by Theorem 3.1 and Theorem 1.1,

we obtain



&u� (t)&u� +(t)&1�|�

t& f9 (u� ({))&1 d{

�|�

t| f $(u)ux | 2 d{�C &u� 0&2

1 |�

t|u({)| p&2

� d{

�C$ |�

t(1+{)&( p&2)�3 d{ � 0

as t � �, since p>5. Using the integral Eq. (3.3), we may write u� + as the form

u� +(t)=S(t)u� 0+|�

0S(t&{) f9 (u� ({)) d{=S(t)w� + , (3.9)

where w� +=u� 0+��0 S(&{) f9 (u� ({)) d{. Since the right side is a linear combina-

tion of solutions of linearized Equation (1.3), u� + is also the solution of (1.3).Now we are going to show the uniqueness of u� +. Let w� + be another

solution of (1.3) which satisfies (1.8). Let V9 (t)#u� +(t)&w� +(t). We want toshows that V9 (t)=0, a.e. Since V9 is also the solution of (1.3), we have

&V9 (0)&1=&S(t) V9 (0)&1=&V9 (t)&1 &u� (t)&w� +(t)&1+&u� (t)&u� +(t)&1 � 0,

as t � �. This implies that &V9 (t)&1=0, for all t�0. Hence u� +=w� +, a.e.This completes the proof of Theorem 1.2.

4. CONSTRUCTION OF SCATTERING OPERATOR S

In this section, we want to construct the scattering operator S (Theorem4.1). It is convenient to rewrite (BQ) in its integral form

u� (t)=S(t) g� +|t

sS(t&{) f9 (u� ({)) d{, (BQs , g� )

where u� is the solution of (BQ) with the initial value u� (s)=S(s) g� at t=s.Formally letting s � \�, we have the integral equations which relate u�to g� \

u� (t)=S(t) g� \+|t

\�S(t&{) f9 (u� ({)) d{. (BQ\� , g� \)

We introduce the norm

&u� &U=supt # R

[(1+|t| )1�3(1&2�( p+1)) &u� (t)&1, p+1+&u� (t)&1] (4.1)

and the space U=[u� # C(R, X p+11 & X1) | &u� &U<�].

60 YUE LIU


Theorem 4.1. Let f # C 1(R) satisfy | f ({)|=O( |{| p) and | f $({)|=O( |{| p&1) as { � 0. Let p> p0#2+- 7. Assume g� & # X1 and S( } ) g� & # U.There exists $>0 such that if &S( } ) g� &&U<$, then there exists a uniquesolution u� of (BQ&� , g� &) in C(R; X p+1

1 & X1) and

&u� (t)&S(t) g� &&1 � 0 (4.2)

as t � &�. Furthermore, there exists a unique g� + # X1 such that

&u� (t)&S(t) g� +&1 � 0 (4.3)

as t � +�. In addition,

&u(t)&21+|v(t)| 2

2&2 |�

&�F(u(t)) dx=&g� &&2

1 (4.4)

and&g� +&1=&g� &&1 , (4.5)

where F $(s)= f (s) with F(0)=0.

Proof. First of all, we are going to show the existence of solution u� for(BQs , g� &) for any &��s�+�. Consider a complete metric subspaceU($1)=[u� # U | &u� &U�$1] of U and a map T

Tu� (t)=S(t) g� &+|t

sS(t&{) F(u� ({)) d{ (4.6)

for &��s��. We now want to apply the contraction principle in U($1).Using Lemma 2.3, we estimate

&Tu� (t)&1, p+1�&S(t) g� &&1, p+1+C } |t

s( |t&{|&1�3(1&2�( p+1))

+|t&{|&1�2(1&2�( p+1))) & f9 (u� ({)&1, q d{}�&S(t) g� &&1, p+1+C } |

t

s( |t&{|&1�3(1&2�( p+1))

+|t&{|&1�2(1&2�( p+1))) |u| p&1p+1 |ux |p+1 d{}

�&S(t) g� &&1, p+1+C } |t

s( |t&{|&1�3(1&2�( p+1))

+|t&{|&1�2(1&2�( p+1))) &u� ({)& p1, p+1 d{} , (4.7)



where 1�q+1��( p+1)=1. Therefore,

(1+|t| )1�3(1&2�p+1) &Tu� (t)&1, p+1

�(1+|t| )1�3(1&2�p+1) &S(t) g� &&1, p+1+CI &u� & pU , (4.8)

where

I= supt, s # R

(1+|t| )1�3(1&2�( p+1)) } |t

s( |t&{| &1�3(1&2�p+1))

+|t&{|&1�2(1&2�( p+1)))(1+|{| )&1�3(1&2�p+1)) p d{} .Since p> p0=2+- 7 we have

1�3(1&2�( p+1))<1�2(1&2�( p+1))<1<1�3(1&2�p+1)) p.

This implies that the integral is O( |t|&1�3(1&2�( p+1))) as |t| � �, and I isbounded uniformly. On the other hand,

&Tu� (t)&1�&S(t) g� &&1+ } |t

s&S(t&{) f9 (u� ({))&1 d{}

�&S(t) g� &&1+C } |t

s| fx(u({))| 2 d{}

�&S(t) g� &&1+C } |t

s&u� ({))&1, p+1 d{}

�&S(t) g� &&1+C |�

&�(1+|{| )&1�3(1&2�( p+1)) p d{ &u� & p

U

�&S(t) g� &&1+C &u� & pU . (4.9)

Combining (4.8) with (4.9), we obtain

&Tu� &U�&S( } ) g� &&U+C(I+1) &u� & pU . (4.10)

If we choose $�$1�2 and $1 so small that C(I+1) $ p&11 �1�2, then

&Tu� &U�$1 for any u� # U($1). That is, T: U($1) � U($1).

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On the other hand, let u� , v� # U($1), u� =(u, u1), v� =(v, v1)

&(Tu� &Tv� )(t)&1, p+1� } |t

s( |t&{|&1�3(1&2�( p+1))

+|t&{|&1�2(1&2�( p+1))) | f $(u)ux& f $(v)vx |q d{}�C } |

t

s( |t&{|&1�3(1&2�( p+1))+|t&{|&1�2(1&2�( p+1)))

_(&u� & p&11, p+1+&v� & p&1

1, p+1) &u� &v� &1, p+1 d{} .Hence

(1+|t| )1�3(1&2�( p+1)) &(Tu� &Tv� )(t)&1, p+1�CI(&u� & p&1U +&v� & p&1

U ) &u� &v� &U .

(4.11)

Now we estimate in X1

&(Tu� &Tv� )(t)&1

�C } |t

s| fx(u)& fx(v)| 2 d{}

�C |�

&�|( |u| p&2+|v| p&2) |u&v| | 2( p+1)�( p&1) |ux |p+1

+|v| p&12( p+1) |ux&vx | p+1 d{

�C |�

&�[|(u&v)({)| � |ux | p+1 ( |u({)| p&2

2( p+1)( p&2)�( p&1)

+|v({)| p&22( p+1)( p&2)�( p&1))+|v({)| p&1

2( p+1) |ux&vx |p+1] d{

�C |�

&�(&u({)& p&1

1, p+1+&v({)& p&11, p+1) &(u&v)({)&1, p+1 d{

�C(&u� & p&1U +&v� & p&1

U ) &u� &v� &U . (4.12)

Combining (4.11) with (4.12), we obtain

&Tu� &Tv� &U�C(I+1)(&u� & p&1U +&v� & p&1

U ) &u� &v� &U . (4.13)



If $1 is chosen so small that 2C(I+1) $ p&11 �1�2, then T is a contradiction

map on U($1) and so it has a unique solution of (BQs , g� 1) in U($1). It iseasy to see u� is a unique solution in U($1). In fact, let v� be another solutionof (BQs , g� &). Then we have

&v� &U�&S( } ) g� &&U+CI &v� & pU�$+CI &v� & p

U .

If we choose $ so small that cI(2$) p&1�1�2, then &v� &U�2$�$1 . By theuniqueness of solution of (BQs , g� &) in U($1), we obtain v� (t)=u� (t), forall t.

Now we are going to show (4.4). We rewrite (BQs , g� &) as

u� s(t)=S(t) u� 0, s+|t

0S(t&{) f9 (u� s({)) d{, (4.14)

where u� 0, s=g� &&�s0 S(&{) f9 (u� s({)) d{. We find

&u� 0, s&1�&g� &&1+|s

0| fx(u({))| 2 d{�&g� &&1+C &u� & p

U�c$1+C$ p1 , (4.15)

where the constant C is independent of s. If $1 is chosen sufficiently small,by the uniqueness of solution of (BQ) (Theorem 3.1), we have

&u� s(t)&21&2 |

�

&�F(us(t)) dx=&u� s(s)&2

1&2 |�

&�F(us(s)) dx

=&S(s) g� &&21&2 |

�

&�F(S(s) g� &) dx. (4.16)

Using (2.12) in Lemma 2.3, we obtain

} |�

&�G(S(s) g� &) dx}

�C |�

&�|S(s) g� &| p+1�C &S(s) g� && p+1

1, p+1

�C( |s|&1�3(1&2�( p+1))+|s| &1�2(1&2�( p+1))) p+1 &g� && p+11, q � 0, (4.17)

as s � \�. Hence

lims � &� \&u� s(t)&2

1&2 |�

&�G(u� s(t)) dx+=&g� &&2

1 . (4.18)

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On the other hand, let u� be the solution of (BQ&� , g� &). Since &u� s&U�$1

for &��s��, and $1 is independent of s, then

&u� s(t)&u� (t)&1� } |t

s| fx(u({))& fx(u({))| 2 d{}+|

s

&�| fx(u({))| 2 d{

�C(&u� s& p&1U +&u� & p&1

U ) &u� s&u� &U

+C |s

&�(1+|{| )&1�3(1&2�( p+1)) p d{ &u� & p

U

�C$ p&11 &u� s&u� &U+C$ p

1 =(s), (4.19)

where =(s)=�s&� (1+|{| )&1�3(1&2�( p+1)) p d{ � 0, as s � &�, and

(1+|t| )1�3(1&2�( p+1)) &u� s(t)&u� (t)&1, p+1

�CI(&u� s& p&1U +&u� & p&1

U ) &u� s&u� &U+C'(s) &u� & pU , (4.20)

where

'(s)=supt # R

(1+|t| )1�3(1&2�( p+1)) |s

&�( |t&{| &1�3(1&2�( p+1))

+|t&{|&1�2(1&2�( p+1)))(1+|{| )&1�3(1&2�( p+1)) p d{, (4.21)

where '(s) � 0 as s � &�. This can be seen by breaking up the integralas before. Hence we obtain

&u� s&u� &U�2C$ p&11 &u� s&u� &U+C$ p

1(=(s)+'(s)). (4.22)

Choosing $1 so small that 2C$ p&11 �1�2, we obtain &u� s&u� &U � 0 as

s � &�. Now we want to estimate |��&� (F(us(t)&F(u(t))) dx|.

} |�

&�(F(us(t)&F(u(t))) dx }�C |

�

&�( |us(t)| p+|u(t)| p |us(t)&u(t)| dx

�C( |us(t)| pp+1+|u(t)| p

p+1) |us(t)&u(t)|p+1

�C(&u� s& pU+&u� & p

U) &u� s&u� &U

�2C$ p1 &u� s&u� &U � 0, (4.23)



as s � &�. Hence, taking the limit in (4.18) yields (4.4). Next we are goingto show asymptotic behavior (4.2). Since u� # U($1) we have

&u� (t)&S(t) g� &&1�|s

&�& f9 (u({))&1 d{

�C |t

&�&u({)& p

1, p+1 d{

�C |t

&�(1+|{| )&1�3(1&2�( p+1)) p d{ &u� & p

U

�C$ p1 |

t

&�(1+|{| )&1�3(1&2�( p+1)) p d{ � 0, (4.24)

as t � &�, since 1�3(1&2�( p+1)) p>1, provided p> p0#2+- 7.Finally, we are going to show the existence of g� + # X1 , (4.3) and (4.5).

We define

g� += g� &+|�

&�S(&{) f9 (u� ({)) d{. (4.25)

Since the solution u� of (BQ&� , g� &) is in U($1), we have

&g� +&1�&g� &&1+C |�

&�(1+|{| )&1�3(1&2�( p+1)) p d{ &u� & p

U

�&g� &&1+2C$ p1 . (4.26)

This implies g� + # X1 by g� & # X1 .To prove (4.3), first of all, we can show &S( } ) g� +&U<$1�2. In fact, by

the formula (4.25), it is easy to show

(1+|t| )1�3(1&2�( p+1)) &S(t) g� +&1, p+1�&S( } ) g� &&U+CI &u� & pU , (4.27)

where I is defined as before. Hence by (4.26) and (4.27), we obtain

&S( } ) g� +&U�&S( } ) g� &&U+C(I+1)$ p1 .

If we choose $<$1�4 and C(I+1) $ p&11 <1�2, then

&S( } ) g� +&U<$1�2. (4.28)

Consider the integral equation

u� s(t)=S(t) g� ++|t

sS(t&{) f9 (u� s({)) d{ (BQs , g� +)

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with &��s� +�. Exactly as in the proof of existence of u� s of(BQs , g� &), we can show that (BQs , g� +) has a unique solution u� s in U($1).We rewrite (BQs , g� +) as

u� s(t)=S(t) u� 0, s+|t

0S(t&{) f9 (u� s)) d{, (4.29)

where u� 0, s=g� +&�s0 S(&{) f9 (u� ({)) d{. Hence exactly following the proof of

(4.4), one can find

&u(t)&21+|v(t)| 2

2&2 |�

&�F(u(t)) dx=&g� +&2

1 , (4.30)

where u� is the solution of (BQ&� , g� &). This implies (4.5). The proof of(4.3) is similar to that of (4.2). This completes the proof of Theorem 4.1.

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Decay and Scattering of Small Solutions of a Generalized Boussinesq Equation

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